1
Convolution Properties
DSP for Scientists
Department of Physics
University of Houston
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Properties of Convolution in Signal Processing, Study notes of Physics
The properties of convolution in the context of digital signal processing (dsp). Topics include the identity, commutative, associative, distributive, and transference properties of convolution. The document also covers special cases such as auto-regression (ar) and moving average (ma) models, and the central limit theorem. Additionally, it discusses the concept of correlation and its application in signal detection.
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Pre 2010
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Convolution Properties
DSP for Scientists Department of PhysicsUniversity of Houston
Properties of Delta Function w
[
n
]:
Identity for Convolution
w^
x
[ n
]^
[
n
] =
x
[ n
]
w^
x
[ n
]^
k
[
n
] =
kx
[ n
]
w^
x
[ n
]^
[
n + s
] =
x
[ n + s
]
Properties of Convolution w^
Associative:{
a
[ n
]^
b
[ n
]}
c
[ n
] =
a
[ n
]^
b
[ n
]^
c
[ n
]}
w^
If w^
a
[ n
]^
b
[ n
]^
c [
n
]^
y
[ n
]
w^
Then w^
a
[ n
]^
b
[ n
]^
c
[ n
]^
y [
n
]
Properties of Convolution w^
Distributive a
[ n
]∗
b
[ n
] +
a
[ n
]∗
c [
n
] =
a
[ n
]∗
b
[ n
] +
c
[ n
]}
If
b
[ n
]
w^
a
[ n
]^
y
[ n
]
c [
n
]
Then w^
a
[ n
]^
b
[ n
]+
c [
n
]^
y
[ n
]
Continue w^
If
x
[ n
]^
h
[ n
]^
y [
n
]
Linear
Same Linear
System
System
Then
x^1
[ n
]^
h
[ n
]^
y
[n] 1
Special Convolution Cases w^
Auto-Regression (AR) Model w^
y [
n
] =
k = 0, M - 1.
h
[ k
] x
[ n
k
]
w^
For Example:
y
[ n
] =
x
[ n
] -
x
[ n
- 1]
w^
(first difference)
Example w^
For One-order Difference Equation (MAModel) w^
y [n] =
ay
[n - 1] +
x
[n]
w^
Find the Impulse Response, if the system is w^
(a) Causal w^
(b) Anti-causal
Causal System Solution w^
Input:
[
n
]^
Output:
h
[ n
]
w^
For Causal system,
h
[ n
] = 0,
n
w^
h
[0] =
ah
[-1] +
[0] = 1
w^
h
[1] =
ah
[0] +
[1] =
a
w^
w^
h
[ n
] =
a
n^ u
[ n
]
Central Limit Theorem w^
If a pulse-like signal is convoluted withitself many times, a Gaussian will beproduced. w^
a
[ n
]^
w^
a
[ n
]^
a
[ n
]^
a
[ n
]^
a
[ n
] = ???
Central Limit Theorem
Correlation Detector
- 1 5^
- 1 0^
- 5^
0
5
1 0^
1 5
- 2 5^1 0. 80. 60. 40. 2^0 - 0. 2- 0. 4- 0. 6- 0. 8- 1
- 2 0^
- 1 5^
- 1 0^
- 5^
0
5
1 0^
1 5^
2 0^
2 5
(^2) 1. 5 (^1) 0. 5 (^0) - 0. 5- 1 - 1. 5- 2
Correlation Results
0
2 0 0
4 0 0
6 0 0
8 0 0
1 0 0 0
1 2 0 0
(^86420) -2 -4 -
-5^ -
-^
-2^ -^
0 1
2 3
4 5
(^1) 0.8 0.6 0.4 0.2 (^0) -0.2 -0.4 -0.
Example: Lowpass
0
50
100
150
200
250
300
350
14012010080 60 40 20 0 -20 -40 -
0
50
100
150
200
250
300
350
14012010080 60 40 20 0 -20-40-
-^ -^ -^ -^ -^
0
2
4
6
8
1 0
0.5 0.4 0.3 0.2 0.1^0 -0.
High-Pass Filter w^
Filter
g
[ n
]:
w^
Remove the Average Value of Signal(Direct Current Components), OnlyPreserve the Quick Undulation Terms w^
n^
g
[ n